1. Border radius? Don't really know, pretty big actually. But the tiles you can work will always be 21.
3. Depends. If you are the founder of a religion, when you generate a GP and settle it in your holy city you get +1gpt for every city that follows your religion. This is actually a veeery big deal. On the other hand if you don't follow the religion of a civ, they will hate you. Welcome to the diplo in IV, where it all comes down to religion and religion only, with only a precious few civs not being influenced by it as much.
So, you can found your own religion and get immediately to work on missionaries to spread it as far as you can (you need open borders to convert other civ's cities). You can wait for another civ to convert you or you can found a religion, convert a civ you don't like, NOT convert to your very own religion, and wait for the other civs to destroy the one you converted. It's just as ridiculous as it sounds.
Duplicate luxuries don't give you extra happiness, no. Trade trade them in for gold or other resources. Note that unlike in V, in IV you sell resources only for gold per turn (so you can't sell your resources just before a war. How they managed to mess this up is beyond me).
4. That's you diplo screen. Right-click on any portrait to contact that leader.
5. Civics, it's a civic. You unlock it by researching Bronze Working (great logic there). Btw if you build the pyramids, always switch to Representation and try to turn your capital into a Great Scientist factory. Why your capital? Because you usually get a high-food start and when you research Civil Service IIRC you unlock the Bureaucracy civic (+50% commerce and production in your capital) Unlike in V scientists boost a city's science output by 50%, which makes them rather valuable.
6. I mean lower the slider to 0
You have the slider, you can lower it to 0 before you get a library up, generate a lot of money and then crank it up to 100%. You will surely be losing money, but you will be researching a lot (thanks to the library). Here money is research. That's about it.
I don't really think it's that much more complicated than V. Maintenance is something you really have to keep a track of early on until you get courthouses up and running (Code of Laws). Local health and happiness aren't as hard to manage as you would think, and health specifically is kind of superficial. Espionage in IV can be quite annoying, watch out for it.
Some completely random advice:
The Pyramids are epic. Pure epic, try to get them by all means.
Spam a lot of units early on to "fog bust". Barbs spawn in the fog of war, so having units everywhere means no barbs. They are
a lot more dangerous in IV, annoyingly so, and will upgrade with time. You haven't seen anything until you fight barbarian riflemen
They can also take cities. At higher difficulty settings they even
found cities.
Initial build order is very different. Almost always start with a worker, followed by lots of warriors to fog bust. Get your first settler out when you reach size 3. Keep expanding as fast as possible without ruining your economy. Having 3-4 cities will probably lose you the game, although I do have an Emperor win with 3 cities.
You don't need to research everything. You can completely bypass Archery and hunting for instance if you don't have any resources that require hunting. Or you can ignore the religious techs if you don't plan on founding a religion. You will need those techs, but you can get them from the AI civs. Tech trading is a great feature, I don't understand why they removed it.
Early on research the techs you need to build improvements on the resources around you and then get bronze working to reveal copper and get slavery. After that it's your choice, I personally love going Mysticism-Polytheism-Priesthood, start building the Oracle and in the meantime research Writing, and with the Oracle get Code of Laws. It's a critical tech that gets you a religion, a very important building (courthouses) and its high value means you can get a lot for it out of the other civs. Oh, and don't forget Pottery, cottages are very important (generate money aka generate beakers) and Granaries are crucial to using the whip correctly.
The AI ignore Alphabet, it never gets it. As soon as you do (it enables tech trading) start trading it in for the techs you did ignore.
Monarchy gives you +1 happiness for every military unit in a city. An easy way to combat it early on, although if you have the Pyramids, Representation is all you need.
You will need at least 8 cities for national wonders. This isn't CiV, where both tall and wide empires are a viable option. You expand or you die. More cities means more production, more commerce and more research. There is no trade-off.
In the late game you will encounter war weariness. When you are at war your population will be unhappy. A lot after a certain point, a whole lot. You can combat this with civics and buildings.
Siege units not only bombard cities, they also deal damage to many units in a stack. So sacrificing several of them to weaken a big stack is a very nice tactic.
The UN can do a lot more than just win you the game. There are different resolutions, including one that bans nuclear weapons. And in vanilla if that one gets a positive vote that's it, no more nuclear weapons.